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month and repeat for several months. How stable is the beta estimate?

You can also calculate betas using monthly data from the Monthly Valuation Data

2.

report in Market Insight. Calculate beta using 2 years of monthly data, and repeat

for different periods. Are the 2-year betas more stable than the one month betas

computed in question 1? What are the trade-offs that an analyst might consider

in using beta computed from long-term versus short-term data sets?

WEBMA STER

Returns, Risk, and Correlations

Go to www.mhhe.com/edumarketinsight. Pull the monthly returns for General Electric,

The Home Depot, Johnson & Johnson, Honeywell, and H.J. Heinz. Merge the returns

for these five firms into a single Excel workbook, with the returns for each company

properly aligned.

Then,

1. Using the Excel functions for average and standard deviation (sample), calculate

the average and standard deviation for each of the firms.

2. Using the correlation function, construct the correlation matrix for the five funds

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based on their monthly returns for the entire period. What are the lowest and

highest individual pair of correlation coefficients?

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 6. Efficient Diversification Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

206 Part TWO Portfolio Theory

SOLUTIONS TO 1. Recalculation of Spreadsheets 6.1, 6.2, and 6.4 shows that the correlation coefficient with the new

rates of return is .98.

>

Concept A B C D E F

Stock Fund Bond Fund

CHECKS 1

Scenario Probability Rate of Return Col. B Col. C Rate of Return Col. B Col. E

2

Recession 0.3 -12 -3.6 10 3

3

Normal 0.4 10 4 7 2.8

4

Boom 0.3 28 8.4 2 0.6

5

Expected or Mean Return SUM: 8.8 SUM: 6.4

6

7

8

Stock Fund Bond Fund

9

Squared Deviations Squared Deviations

10

Scenario Probability from Mean Col. B Col. C from Mean Col. B Col. E

11

Recession 0.3 432.64 129.792 12.96 3.888

12

Normal 0.4 1.44 0.576 0.36 0.144

13

Boom 0.3 368.64 110.592 19.36 5.808

14

SUM: SUM:

Variance = 240.96 9.84

15

Std Dev = Variance 15.52 3.14

16

17

Deviation from Mean Return Covariance

18

Scenario Probability Stock Fund Bond Fund Product of Dev Col. B Col. E

19

Recession 0.3 -20.8 3.6 -74.88 -22.464

20

Normal 0.4 1.2 0.6 0.72 0.288

21

Boom 0.3 19.2 -4.4 -84.48 -25.344

22

Covariance: SUM: -47.52

23

Correlation coefficient = Covariance/(StdDev(stocks)*StdDev(bonds)): -0.98

24

2. a. Using Equation 6.3 with the data: 12; 25; wB 0.5; and wS 1 wB 0.5, we

B S

obtain the equation

2

152 2 2

(wB B) (wS S) 2(wB B)(wS S) BS

P

12)2 25)2

(0.5 (0.5 2(0.5 12)(0.5 25) BS

which yields 0.2183.

b. Using Equation 6.2 and the additional data: E(rB) 6; E(rS) 10, we obtain

E(rP) wBE(rB) wSE(rS) (0.5 6) (0.5 10) 8%

c. On the one hand, you should be happier with a correlation of 0.2183 than with 0.22 since the

lower correlation implies greater benefits from diversification and means that, for any level of

expected return, there will be lower risk. On the other hand, the constraint that you must hold

50% of the portfolio in bonds represents a cost to you since it prevents you from choosing the

risk-return trade-off most suited to your tastes. Unless you would choose to hold about 50% of

the portfolio in bonds anyway, you are better off with the slightly higher correlation but with the

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ability to choose your own portfolio weights.

3. The scatter diagrams for pairs Bâ€“E are shown below. Scatter diagram A shows an exact conflict

between the pattern of points 1,2,3 versus 3,4,5. Therefore the correlation coefficient is zero.

Scatter diagram B shows perfect positive correlation (1.0). Similarly, C shows perfect negative

correlation ( 1.0). Now compare the scatters of D and E. Both show a general positive correlation,

but scatter D is tighter. Therefore D is associated with a correlation of about .5 (use a spreadsheet to

show that the exact correlation is .54) and E is associated with a correlation of about .2 (show that

the exact correlation coefficient is .23).

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 6. Efficient Diversification Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

207

6 Efficient Diversification

Scatter diagram C

Scatter diagram B

6

6

5

5

4

4

Stock 2

Stock 2

3

3

2

2

1

1

0

0

0 1 2 3 4 5 6

0 1 2 3 4 5 6

Stock 1

Stock 1

Scatter diagram E

Scatter diagram D

6

6

5

5

4

4

Stock 2

Stock 2

3

3

2

2

1

1

0

0

0 1 2 3 4 5 6

0 1 2 3 4 5 6

Stock 1

Stock 1

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Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 6. Efficient Diversification Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

208 Part TWO Portfolio Theory

4. a. Implementing Equations 6.2 and 6.3 we generate data for the graph. See the spreadsheet above.

b. Implementing the formulas indicated in the following spreadsheet we generate the optimal risky

portfolio (O) and the minimum variance portfolio.

c. The slope of the CAL is equal to the risk premium of the optimal risky portfolio divided by its

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standard deviation, (11.28 5)/17.59 .357.

d. The mean of the complete portfolio .2222 11.28 .7778 5 6.40% and its standard

deviation is .2222 17.58 3.91%.

The composition of the complete portfolio is

.2222 .26 .06 (i.e., 6%) in X

.2222 .74 .16 (i.e., 16%) in M

and 78% in T-bills.

5. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on

various securities and estimates of risk, that is, standard deviations and correlation coefficients.

The forecasts themselves do not control outcomes. Thus, to prefer a manager with a rosier forecast

(northwesterly frontier) is tantamount to rewarding the bearers of good news and punishing the

Bodieâˆ’Kaneâˆ’Marcus: II. Portfolio Theory 6. Efficient Diversification Â© The McGrawâˆ’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

209

6 Efficient Diversification

Optimal risky portfolio

25

CAL

20

Portfolio mean (%)

15 Efficient frontier of

risky assets

O X

Min. Var. Pf

10

M

5 C

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